Chapter P, Problem 15E

The Rule of 72 This is a continuation of Exercise 14. Financial advisors sometimes use a rule of thumb known as Rule of 72 to get a rough estimate of the time it takes for an investment to double in value. For an investment that is compounded yearly at an interest rate of $r\%$, this rule says it will take about $72/r$ years for the investment to double. In this calculation, $r$ is the integer interest rate rather than a decimal. Thus, if the interest rate is $8\%$, we would use $72/8$ rather than $72/0.08$.

For the remainder of this exercise, we will consider an investment that is compounded yearly at an interest rate of $13\%$.

a. According to the Rule 72, how long will it take the investment to double in value?

Parts b and c of this exercise will check to see how accurate this estimate is for this particular case.

b. Using the answer you got from part a of this exercise, calculate the future value interest factor (as defined in Exercise 14). Is it exactly the same as your answer to the part a of Exercise 14?

c. If your initial investment was $\$5000$, use your answer from part b to calculate the future value. Did your investment exactly double?

Future Value Business and finance texts refer to the value of an investment at a future time as its future value. If an investment of P dollars is compounded yearly at an interest rate of $r$ as a decimal, then the value of the investment after $t$ years is given by

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=P\times {\left(1+r\right)}^t$.

In this formula, ${\left(1+r\right)}^t$ is known as the future value interest factor, so the formula above can be written as

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=P\times \mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{ }\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{ }\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}$

Financial officers normally calculate this (or look it up in a table)

a. What future value interest factor will make an investment double?

b. Say you have an investment that is compounded yearly at a rate of $9\%$.

Find the future value interest factor for a 7-year investment.

c. Use the results from part b to calculate the 7-year future value if your initial investment is $\$5000$.